Paper by Erik D. Demaine
- Robert Connelly, Erik D. Demaine, and Günter Rote, “Infinitesimally Locked Linkages with Applications to Locked Trees”, in Abstracts from the 11th Annual Fall Workshop on Computational Geometry, New York, New York, November 2–3, 2001.
Recently there has been much interest in linkages (bar-and-joint frameworks)
that are locked or stuck in the sense that they cannot be moved
into some other configuration while preserving the bar lengths and not crossing
any bars. We propose a new algorithmic approach for analyzing whether planar
linkages are locked in many cases of interest. The idea is to examine
degenerate frameworks in which multiple edges converge to geometrically
overlapping configurations. We show how to study whether such frameworks are
locked using techniques from rigidity theory, in particular first-order
rigidity and equilibrium stresses. Then we show how to relate locked
degenerate frameworks to locked frameworks that closely approximate the
degenerate frameworks. Our motivation is that most existing approaches to
locked linkages are based on approximations to degenerate frameworks. In
particular, we show that a previously proposed locked tree in the
can be easily proved locked using our techniques, instead of the tedious
arguments required by standard analysis. We also present a new locked tree in
the plane with only one degree-3 vertex and all other vertices degree 1 or 2.
This tree can also be easily proved locked with our methods, and implies that
the result about opening polygonal arcs and cycles 
is the best possible.
- The abstract is 2 pages.
- The abstract is available in PostScript (115k) and gzipped PostScript (42k).
- See information on file formats.
- [Google Scholar search]
- Related papers:
- InfinitesimallyLocked_LasVegas (Infinitesimally Locked Self-Touching Linkages with Applications to Locked Trees)
See also other papers by Erik Demaine.
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